Initial Covariance Matrix Research Articles

Output Variance–Constrained LQG Control of Discrete-Time Systems

The constrained infinite-horizon LQG control problem can be solved via semidefinite programming if the state-control constraints are given by variance-bounds on linear functions of the state and control input. Given a nonzero initial state covariance matrix, each suboptimal dynamic output feedback controller is initially time-varying but reaches time-invariance after a finite number of time steps. This number of time steps can be determined iteratively via repetitive execution of semidefinite programs.

Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data

The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness.

Adaptive radial-based direction sampling: some flexible and robust Monte Carlo integration methods

Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with non-elliptical, possibly, multimodal target distributions. A key step is a radial-based transformation to directions and distances. After the transformation a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. The ARDS algorithms are illustrated on a regression model with scale contamination and a mixture model for economic growth of the USA.

On asymptotic stability of continuous-time risk-sensitive filters with respect to initial conditions

In this paper, we consider the problem of risk-sensitive filtering for continuous-time stochastic linear Gaussian time-invariant systems. In particular, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaussian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial conditions in the mean square sense at an exponential rate, provided the arbitrary initial covariance matrix results in a stabilizing solution of the ( H ∞-like) Riccati equation associated with the risk-sensitive problem. More importantly, in the case of non-Gaussian initial conditions, a suboptimal risk-sensitive filter asymptotically approaches the optimal risk-sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon–Nikodym derivative.

Parametric Estimation: Improvement Of The Rls Algorithm Using A Differential Approach

This paper describes a new computational method for recursive least squares (RLS) algorithm. It is well known that the initial values for computing RLS estimates should be chosen to guarantee the existence of the estimates at each step, that the initial covariance matrix may affect the convergence rate, and that a blow-up phenomenon (infinite increase of the covariance matrix) can appear. Much research has focused on each of these problems, but heavy computations and different specific design parameters result from the most common solutions. We propose an alternative simple algorithm that reaches a trade-off between the advantages of the well-known RLS algorithms and of the more complex computations. The proposed algorithm modifies the prior additional term used in the cost function. This modified term is updated during the recursion, in the context of a differential formalism. The aim of this formalism is, first, to provide valid computations and properties in both discrete and continuous time domains, and to yield the variations of the parameters at each step. Second, in the particular case of discrete time this formalism provides very good numerical properties. The methodology in developing this approach is quite different from the previous work on RLS estimation. We demonstrate that our algorithm, compared with RLS algorithm, leads to a higher convergence rate. It also offers a closer tracking of parameters in nonstationary case and satisfactory results in the presence of blow-up situations. This goal is achieved without significant additional computations, and the behaviour of the approach is illustrated through simulations.

Influence of the Initial Covariance Matrix on Recursive LS Estimation of Continuous Models via Generalised Poisson Moment Functionals

This paper presents the results of an investigation on the quality of recursive parameter estimation via the ordinary and normalised Poisson moment functional approach. It is shown that the ordinary and the normalised approach have different requirements for the initial value of the covariance matrix in the recursive least-squares estimation algorithm. Correctly initialised, both approaches supply the same results, letting aside numerical problems.

Cuticular Hydrocarbon Discrimination of Diabrotica (Coleoptera: Chrysomelidae) Sibling Species

Cuticular hydrocarbons were extracted from adults of the sibling species Diabrotica longicornis (Say) and D. barberi Smith & Lawrence and characterized by gas chromatography (GC) and GC-mass spectrometry (GC-MS). Six classes of hydrocarbons were present: n -alkanes, terminally branched monomethylalkanes, internally branched monomethylalkanes, dimethylalkanes, n -alkenes, and monomethyl-branched alkenes. Diet and age did not significantly affect cuticular hydrocarbon patterns in most cases. However, when aged beetles were analyzed, the quantity of several cuticular hydrocarbons changed after the teneral stage. Some sexual dimorphism was seen in cuticular hydrocarbons of older beetles, but it did not significantly affect comparisons between species. Identical classification results were obtained when stepwise discriminant analysis using 13 cuticular hydrocarbons as variables was compared with visual identification of 60 reared beetles (34 D. longicornis and 26 D. barberi from allopatric populations). When additional reared and field- collected beetle profiles (from sympatric and allopatric populations) were entered into the discriminate analysis using the initial covariance matrix as a standard, there was 95% agreement with visual identification. These data suggest that, in most cases, the cuticular hydrocarbon analysis may be used to differentiate the sibling species regardless of their origin.

Self-Tuning Adaptive Control of a DC Motor Driven Single Axis Table

Abstract This paper describes the application of self-tuning adaptive control on a laboratory model of a DC motor driven uniaxial table. The effects of forgetting factor, initial estimates and initial covariance matrix on the estimation procedure and the control algorithm are investigated. It was observed that the self-tuning adaptive control provided better response than the fixed parameter control approach.

Cyclomonotonicity and stabilizability properties of solutions of the difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

RLS parameter convergence with overparametized models

This paper, through the concept of quotient space, generalizes the RLS parameter convergence results obtained previously using the Lyapunov type function approach [14,1,12,7] to overparameterized systems. The system signals do not have to be quasi-stationary as assumed by Ljung [10] and a priori knowledge of the degree of overparameterization is not required. The excitation condition is the weakest possible and it is shown that under this excitation condition limiting set of parameter estimates always belongs to a set H, each element of which contains the irreducible plant model plus a monic common factor polynomial. It is further shown that the common factor polynomials defined by the limiting set of parameter estimates are quantitatively related to the initial conditions of the algorithm. For example, if the condition number of the initial covariance matrix is unity, then the limiting set of parameter estimates is the singleton on H closest to the initial parameter estimate.

A NOTE ON THE GENERATION OF INDEPENDENT REALIZATIONS OF A VECTOR AUTOREGRESSIVE MOVING‐AVERAGE PROCESS

Abstract. Barone has described a method for generating independent realizations of a vector autoregressive moving‐average (ARMA) process which involves recasting the ARMA model in state space form. We discuss a direct method of computing the initial state covariance matrix T0 which, unless the number of time series is large, is usually faster than using the doubling algorithm of Anderson and Moore. Our numerical comparisons are particularly valuable because T0 must also be computed when calculating the likelihood function. A number of other computational refinements are described. In particular, we advocate the use of Choleski factorizations rather than spectral decompositions. For a pure moving‐average process computational savings can be achieved by working directly with the ARMA model rather than with its state space representation.

The determination of the state covariance matrix of moving-average processes without computation

It is shown that by choosing a suitable state space representation of an MA( q) process the necessity to compute the initial state covariance matrix does not arise. This result reduces the computational efforts for maximum likelihood estimation via Kalman filtering.

The exact initial covariance matrix of the state vector of a general MA(q) process

Abstract The finite moving-average process is studied in state-space form, and its stochastic structure is exploited to derive a closed-form analytic expression for the covariance matrix of the initial state vector. The results provide interesting insight into the state-space structure of moving-average processes, and they minimize the computational requirements for exact maximum-likelihood estimation via the Kalman filter.

Monotonicity and stabilizability- properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples

The problem considered is that of selecting an initial covariance matrix for the Kalman filter to ensure that the closed-loop filter at every subsequent time instant is exponentially asymptotically stable as a time-invariant filter. Sufficient conditions are derived based on monotonicity properties of the solution of the Riccati difference equation. The results have application in observer design, and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included.

On complementary models and fixed-interval smoothing

A new algorithm is derived for the standard fixed-interval linear smoothing problem in which the signal is generated by a state model. The structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix \Pi_{0} , since, unlike in existing algorifiuns, the relevant Riccati equation is entirely independent of \Pi_{0} . The derivation of the algorithm is based on properties of complementary models.

Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations

The method of calculating the exact likelihood function of a stationary autoregressive moving average (ARMA) time series based on Akaike's Markovian representation and using Kalman recursive estimation is reviewed. This state space approach involves matrices and vectors with dimensions equal to Max (p, q + 1) where p is the order of the autoregression and q is the order of the moving average, rather than matrices with dimensions equal to the number of observations. A key to the calculation of the exact likelihood function is the proper calculation of the initial state covariance matrix. The inclusion of observational error into the model is discussed as is the extension to missing observations. The use of a nonlinear optimization program gives the maximum likelihood estimates of the parameters and allows for model identification based on Akaike's Information Criterion (AIC). An example is presented fitting models to western United States drought data.

Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations

The method of calculating the exact likelihood function of a stationary autoregressive moving average (ARMA) time series based on Akaike's Markovian representation and using Kalman recursive estimation is reviewed. This state space approach involves matrices and vectors with dimensions equal to Max (p, q + 1) where p is the order of the autoregression and q is the order of the moving average, rather than matrices with dimensions equal to the number of observations. A key to the calculation of the exact likelihood function is the proper calculation of the initial state covariance matrix. The inclusion of observational error into the model is discussed as is the extension to missing observations. The use of a nonlinear optimization program gives the maximum likelihood estimates of the parameters and allows for model identification based on Akaike's Information Criterion (AIC). An example is presented fitting models to western United States drought data.

Suboptimal linear regulators with incomplete state feedback

A method of designing linear regulators with incomplete state feedback has been suggested by Rekasius [1]. Ramar and Ramaswami [2] have pointed out the difficulties encountered in applying this method. This correspondence presents, briefly, an alternative approach to this problem in two cases of a) unknown initial state and b) known initial state statistics, viz., mean and covariance matrix. Solution for the control law utilizing only the available states is obtained by minimizing an upper bound on the ratio of the suboptimal to optimal cost in case a). In case b) the expected value of the suboptimal cost is minimized. It is assumed that the available states are sufficient to make the feedback system stable. The solution is in the form of necessary conditions and results in a set of simultaneous polynomial equations, but the solution to the optimal control problem is not required.